Another look at a priori relationships among traffic flow characteristics. Discussions. Author's closures

Past derivations of a priori relationships among speed, flow, and concentration (such as the fundamental relationship and the speed-flow-occupancy relationship) have involved unrealistic assumptions of uniformity in at least one traffic flow characteristic. Several relationships are derived for which these assumptions of uniformity are relaxed. For relationships involving both time- and distance-based variables, this requires that the relationship be understood in probabilistic terms ; where all variables are time-based, deterministic relationships are also possible. The fundamental relationship can be shown to be strictly true in the limit where the time and distance intervals over which measurements are taken approach 0. Where the order of arrival of vehicles with particular speeds is random, the fundamental relationship is found to hold for average values of the variables in question ; this is also true if the section over which density is measured is empty at the beginning and end of the time interval used for averaging. Relationships derived under various other assumptions involve covariance terms, so that if particular variables are not correlated, simple relationships continue to hold. Where these variables are correlated, biases may be expected. Under certain conditions, these biases may be quite serious. Comparison of the relationships derived here with those of past empirical studies results in good agreement for the relationship between density, as estimated from the fundamental relationship, and occupancy. On the other hand, previously reported discrepancies between measured speeds and speeds calculated from flows and occupancies cannot be explained fully by the covariance terms in the relationships derived here.